A Commodity-Money Refinement in
Matching Models
Neil Wallace∗
Department of Economics
The Pennsylvania State University
608 Kern
University Park, Pa., 16802
[email protected];
Tao Zhu
Department of Economics
Cornell University
442 Uris Hall
Ithaca, NY 14853
[email protected]
August 26, 2003
∗
Corresponding author: Neil Wallace, Department of Economics, The Pennsylvania
State University, 608 Kern, University Park, Pa., 16802; email: [email protected]; tel 814863-3805; fax 814-863-4775.
1
Abstract
We apply a commodity-money refinement to matching models in which
people meet in pairs and buyers make take-it-or-leave-it offers to sellers. The
refinement is applied by attaching a utility value to nominal money and
letting that value approach zero. An equilibrium satisfies the refinement if
it is such a limit. We show that the refinement eliminates a class of non
full-support steady states.
J. of Econ. Lit. Classification E40.
Running title: Commodity-Money Refinement
2
1
Introduction
The conception of money as fiat money, an intrinsically useless object, has
been used in models for a long time. It was used in the classical-dichotomy
model, even though that model was developed and used at a time when actual
money was gold or silver. Ignoring the commodity aspect of money was
convenient because it produced a simple and strong prediction: allocations
are independent of the amount of money. However, if actual money is, in
fact, a commodity–albeit a “low value” commodity usable, for example, only
as wall-paper or fuel–then we should take seriously only those equilibria
which are limits of commodity-money equilibria as the commodity “value”
approaches zero. Here, we describe some implications of this refinement
(selection device) in some models in which trade occurs between pairs of
people and in which the buyer (of goods) makes a take-it-or-leave-it offer to
the seller. As we will see, the refinement eliminates a class of non full-support
steady states.
We deal with two fairly familiar background models. One is a model in
which people are endowed at alternating dates, a special case of Bewley [1].
In our version, at each date each person with an endowment, a seller, meets
a random person without an endowment, a buyer. The other model is an
indivisible-money version of the random-matching model first set out by Shi
[2] and by Trejos and Wright [3], a multi-good model with specialization in
consumption and production and no double coincidences. Each model has
non full-support monetary steady states that do not satisfy the refinement.
In the first model, even though there is no obvious source of heterogeneity, a
steady state with the degenerate distributions of money holdings for buyers
and sellers that are consistent with competitive trade does not satisfy the
refinement. In the second model, the refinement produces a strengthened
non-neutrality result.
The idea of applying a commodity-money refinement to fiat-money equilibria is not new. Its consequences in overlapping generations models are
well-known. Some such models and others have a class of competitive equilibria in which the value of a fixed stock of fiat money converges to zero.
The refinement is known to eliminate those. It also eliminates non-monetary
equilibria unless those are the only equilibria.
3
2
The Commodity “Value” of Money
There are two main ways to model the commodity value of money. One is
money as jewelry: wear it to get some utility and then keep it or trade it. The
other is money as fuel: use it to get some utility and lose it. Here, we adopt
the fuel specification, although most of the results can be obtained from either
specification. In particular, we let the utility payoff from using an amount of
money m be given by the function γ(m), where ≥ 0 is a “productivity-ofmoney” parameter. We assume that γ : R+ → R+ , is differentiable, bounded,
strictly increasing, and concave, and satisfies γ(0) = 0 and γ 0 (0) finite.
Our models are discrete-time models with one pairwise meeting per date.
Each person enters a date with some money and then meets someone. Each
person can decide how to use his money. In general, it is allocated among
the amounts traded, held, and used up. All our claims are about steady
states in which, by definition, no money is used up. A steady state is defined
below for each model studied. A fiat-money ( = 0) steady state satisfies
the commodity-money refinement if it is a limit of commodity-money steady
states as → 0, where an appropriate limit is defined below for each model.
3
Alternating Endowments with Divisible Money
There is one perishable and divisible good per date. There are two equally
sized intervals of people, which we label group 1 and group 2. Each member
of group 1 has an income stream in the form of goods given by (ω, 0, ω, 0, ...),
while each member of group 2 has an income stream in the form of goods
given by (0, ω, 0, ω, ...), where ω > 0. Everyone has the same preferences
represented by expected discounted utility with discount factor β ∈ (0, 1)
and period utility function u : R+ → R. The function u is strictly increasing,
strictly concave, differentiable, and, without loss of generality, u(0) = 0.
In addition, we assume a lower bound on marginal utility at zero: βu0 (0)
> 2u0 (ω).
There exists a fixed stock of divisible money. At each date a person with
endowment 0, the buyer, meets a person with endowment ω, the seller. Each
trading partner sees the money of his or her trading partner, but any other
information about the partner’s trading history is private.1
1
If there is centralized competitive trade at each date that is limited to trading money
for the good, then there is a steady state in which every person with endowment 0 starts
4
A steady state is a pair of value functions defined on the amount of money
held and a pair of measures that describe the distribution of money holdings,
both of which pertain to the beginning of a date prior to meetings. We let
v0 denote the value function for someone with endowment 0 and vω denote
that for someone with endowment ω and we let π 0 and π ω be the respective
measures. To describe steady-state conditions, it is helpful to first set out
the take-it-or-leave-it choice problem of a buyer with m0 who meets a seller
with mω .
Problem 1 Choose (c, z0 , m) ∈ R3+ to maximize u(c) + γ(z0 ) + βvω (m)
subject to c ≤ ω, z0 + m ≤ m0 , and
u(ω − c) + max[βv0 (mω + m0 − m − z0 − zω ) + γ(zω )]
zω
≥ max[u(ω) + βv0 (mω − z) + γ(z)].
z
(1)
In this problem, the z variables represent the amounts of money used up.
To save space, in (1) we have omitted the obvious non negativity constraints
that limit the choices of zω and z. Let g0 (m0 , mω ) be the maximized objective
and let gω (m0 , mω ) be the implied value of the left-hand side of (1). Also, let
P (m0 , mω ) be the set of maximizers for m.
The steady state must satisfy
Z
Z
v0 (m0 ) = g0 (m0 , mω )dπω (mω ), vω (mω ) = gω (m0 , mω )dπ 0 (m0 ). (2)
Also, let p(., .) : R2+ → R+ be a selection of P (., .) that is Borel measurable
and let the set-valued mappings ζ 0 and ζ ω on the Borel sets of R+ be defined
by
ζ 0 [0, m] = {(m0 , mω ) ∈ R2+ , mω + m0 − p(m0 , mω ) ≤ m},
(3)
ζ ω [0, m] = {(m0 , mω ) ∈ R2+ , p(m0 , mω ) ≤ m}.
(4)
the period with the same amount of money and every person with endowment ω starts
with no money. As is well-known, the steady state consumptions are given by the solution
to βu0 (c) = u0 (ω − c), where c is the consumption of the person with endowment 0. In
this steady state, all money changes hands at each date. It is straightforward to show that
this equilibrium satisfies the refinement.
5
Let π denote the product Borel measure π 0 ×π ω on R2+ . Then, a steady state
must also satisfy
Z
Z
1 =
mdπ0 (m) + mdπω (m),
(5)
π0 [0, m] = π(ζ 0 [0, m]), π ω [0, m] = π(ζ ω [0, m]),
(6)
where, in (5), we have set the mean holding of money to be one half.
That is, we say that (v0 , vω , π 0 , π ω ) is a steady state if it satisfies (2)-(6),
Also, we say that a steady state is a degenerate steady state if π 0 ({1}) = 1
and πω ({0}) = 1. And we say that an = 0 (fiat) steady state is a monetary
steady state if trade occurs in some meetings.
3.1
A degenerate fiat monetary steady state
Here we assume that = 0. With no commodity “value” attached to money,
problem 1 simplifies to
Problem 2 Choose (c, m0 ) ∈ [0, ω] × [0, m0 ] to maximize u(c) + βvω (m0 )
subject to
u(ω − c) + βv0 (mω + m0 − m0 ) ≥ u(ω) + βv0 (mω ).
(7)
If (c, m0 ) solves problem 2, then either c < ω or c = ω. If c < ω, then (7)
holds at equality. In this case, u(ω) − β[v0 (mω + m0 − m0 ) − v0 (mω )] ≥ 0 and
we can solve the constraint at equality for c obtaining
c = f [v0 (mω + m0 − m0 ) − v0 (mω )],
(8)
where f : [0, u(ω)/β] → [0, ω] is defined by
f (x) = ω − u−1 [u(ω) − βx].
(9)
(Notice that f is strictly increasing and strictly concave with f(0) = 0 and
f 0 (0) = β/u0 (ω).) If c = ω, then u(ω)− β[v0 (mω + m0 − m0 ) − v0 (mω )] ≤ 0.
In any case, we conclude that
⎧
⎨ ω if u(ω) ≤ β[v0 (mω + m0 − m0 ) − v0 (mω )]
.
(10)
c=
⎩
0
f [v0 (mω + m0 − m ) − v0 (mω )] otherwise
6
In addition, the following notation is useful. Let c∗ ∈ (0, ω) satisfy
u(ω) = βu(c∗ ) + u(ω − c∗ ).
(11)
(Because x 7→ h(x) ≡ βu(x) + u(ω − x) is strictly concave on [0, ω], h(0) =
u(ω), h(ω) < u(ω), and h0 (0) = βu0 (0) − u0 (ω) > 0, c∗ exists and is unique.)
The first proposition gives some properties of a degenerate monetary
steady state.
Proposition 1 If (v0 , vω , π 0 , π ω ) is a degenerate monetary steady state, then
(i) v0 (0) = βvω (0); (ii) v0 (1) − v0 (0) = u(c∗ ); (iii) vω (0) = u(ω)/(1 − β 2 );
(iv) v0 (m) = v0 (0) for m ∈ (0, 1); (v) vω (m) = vω (0) for m ∈ (0, 1).
Proof. (i) This follows from problem 2.
(ii) By the definition of a degenerate monetary steady state, there must
be (c, m0 ) with c > 0 and m0 = 0 that solves problem 2 for (m0 , mω ) = (1, 0).
It follows that v0 (1) = u(c) + βvω (0), and, hence, from part (i) that
v0 (1) − v0 (0) = u(c).
(12)
If c = ω, then v0 (1) − v0 (0) = u(ω). But by (7), β[v0 (1) − v0 (0)] ≥ u(ω), a
contradiction. So c < ω and (7) holds at equality. Then,
β[v0 (1) − v0 (0)] = u(ω) − u(ω − c).
(13)
It follows from (12) and (13) that βu(c) = u(ω) − u(ω − c). But then c > 0
and the definition of c∗ imply c = c∗ .
(iii) Let (c, m0 ) solve problem 2 for (m0 , mω ) = (1, 0). As just established,
(7) holds at equality. Hence, vω (0) = u(ω) + βv0 (0). This and part (i) imply
the result.
(iv) Let m ∈ (0, 1). Because a buyer with m can offer m to a seller
with 0, v0 (m) ≥ u(c) + βvω (0), where c is the maximum amount of the good
obtained by offering m. Also, v0 (m) < v0 (1); otherwise, a buyer with 1 will
not offer 1 to a seller with 0. It follows that v0 (m) − v0 (0) < v0 (1) − v0 (0),
and, therefore, that c = f[v0 (m) − v0 (0)]. This, v0 (m) ≥ u(c) + βvω (0), and
part (i) imply
v0 (m) − v0 (0) ≥ u[f (v0 (m) − v0 (0))].
(14)
However, by part (ii),
v0 (1) − v0 (0) = u[f (v0 (1) − v0 (0))].
7
(15)
Because (14) has the form x ≥ u[f (x)] and (15) has the form x = u[f(x)]
and because u0 (0)f 0 (0) = βu0 (0)/u0 (ω) > 1, it follows that any solution to
(14) satisfies either v0 (m) = v0 (0) or v0 (m) ≥ v0 (1). Because the latter has
been ruled out, it follows that v0 (m) = v0 (0).
(v) Let m ∈ (0, 1). We have βvω (0) = v0 (0) = v0 (m) ≥ βvω (m) ≥ βvω (0),
where the penultimate inequality follows because the buyer can choose m0 =
m and the last follows from free disposal. Therefore, βvω (0) = βvω (m), as
required.
The next proposition establishes existence of a degenerate monetary steady
state.
Proposition 2 There exists a degenerate monetary steady state.
Proof. Let π ∗0 and π ∗ω be such that π∗0 ({1}) = 1 and π∗ω ({0}) = 1. We
show that there exist value functions v0∗ and vω∗ consistent with π ∗0 and π ∗ω .
The value functions are step functions with jumps at the integers. The proof
has two parts: first, we use a fixed point argument to produce the candidates
for v0∗ and vω∗ ; then we show that they and π ∗0 and π∗ω constitute a steady
state. Because we are constructing step functions with jumps at the integers,
the functions are determined by their values at the integers.
In what follows, i denotes an integer. Let v0∗ (1) − v0∗ (0) ≡ D, where
v0∗ (0) and v0∗ (1) are given by proposition 1, and let R∞
+ denote the set of
nonnegative sequences. Let
∗
∗
V0 = {v0 ∈ R∞
+ : v0 (0) = v0 (0), v0 (1) = v0 (1), v0 (i) − v0 (i − 1) ∈ [0, D]}
and let
Vω = {vω ∈ R∞
+ : vω (i) = u(ω) + βv0 (i) for some v0 ∈ V0 }.
∞
Let the mapping Φ = (Φ0 , Φω ) : V0 × Vω → R∞
+ × R+ be defined by
Φ0 (v0 , vω )(i) =
max
[u(c) + βvω (m0 )]
(c,m0 )∈[0,ω]×{0,...,ω}
(16)
subject to β[v0 (i − m0 ) − v0 (0)] ≥ u(ω) − u(ω − c),
and by
Φω (v0 , vω )(i) = u(ω) + βΦ0 (v0 , vω )(i).
8
Let R∞
+ be equipped with the topology of pointwise convergence. Then
V0 ×Vω is compact. Also, V0 ×Vω is convex. And by the Theorem of Maximum, Φ0 is continuous. It follows that Φω is continuous. If Φ(v0 , vω ) ∈
V0 × Vω , then it follows immediately that Φ has a fixed point. To show that
Φ(v0 , vω ) ∈ V0 × Vω , it suffices to show that Φ0 (v0 , vω ) ∈ V0 .
It is clear that Φ0 (v0 , vω )(0) = v0∗ (0). Now consider Φ0 (v0 , vω )(1). The
choice m0 = 0 implies a payoff of u(c∗ ) +βvω (0), while m0 = 1 implies βvω (1).
By proposition 1, the former is strictly better than the latter. Therefore,
Φ0 (v0 , vω )(1) = v0∗ (1). It remains to show that Φ0 (v0 , vω )(i)− Φ0 (v0 , vω )(i−1)
= δ ∈ [0, D] for i ≥ 2. It is clear that δ ≥ 0. Now let Φ0 (v0 , vω )(i) =
u(c) + βvω (m0 ). Either m0 = 0 or m0 > 0. If m0 > 0, then a lower bound on
Φ0 (v0 , vω )(i − 1) is obtained by evaluating the objective in (16) at (c, m0 − 1).
This gives Φ0 (v0 , vω )(i − 1) ≥ u(c) + βvω (m0 − 1). Then δ ≤ β[vω (m0 ) −
vω (m0 − 1)] ≤ β 2 D. If m0 = 0, there are two possible cases.
Case 1: u(ω) ≤ β[v0 (i) − v0 (0)]. Then c = ω, and a lower bound on
Φ0 (v0 , vω )(i − 1) can be obtained by evaluating the objective in (16) at (c0 , 0)
for some c0 . If u(ω) ≤ β[v0 (i − 1) − v0 (0)], then c0 = ω, and δ ≤ 0. Otherwise,
u(ω) − u(ω − c0 ) = β[v0 (i −1) −v0 (0)]. This and u(ω) ≤ β[v0 (i) − v0 (0)] imply
u(ω − c0 ) ≤ β[v0 (i) − v0 (i − 1)]. It follows that δ ≤ u(ω) − u(c0 ) < u(ω − c0 ) <
D.
Case 2: u(ω) > β[v0 (i) − v0 (0)]. Then c = f [v0 (i) − v0 (0)], and a lower
bound on Φ0 (v0 , vω )(i − 1) can be obtained by evaluating the objective in
(16) at (f[v0 (i − 1) − v0 (0)], 0). It follows that
δ ≤ u[f(v0 (i) − v0 (0))] − u[f (v0 (i − 1) − v0 (0))]
≤ u[f(v0 (i) − v0 (i − 1))] ≤ u[f(D)] = D,
where the second inequality follows from concavity of u and f.
A fixed point of Φ gives values of v0∗ and vω∗ at the integers, and, therefore,
determines the step functions. To complete the proof, we show that the step
functions v0∗ and vω∗ and π ∗0 and π ∗ω constitute a steady state. First, we claim
that
v0∗ (m) =
max
[u(c) + βvω∗ (m0 )]
(17)
0
(c,m )∈[0,ω]×[0,m]
subject to
β[v0∗ (m − m0 ) − v0∗ (0)] ≥ u(ω) − u(ω − c)}.
9
Because v0∗ and vω∗ are step functions, the constraint m0 ∈ [0, m] in (17) can
be replaced by m−m0 ∈ {0, 1, ..., int(m)}, where int(m) is the largest integer
no greater than m. Then, for m = i, the claim holds because v0∗ and vω∗ are
derived from a fixed point of Φ. For m ∈ [i, i + 1), the claim follows from the
result for integers and the fact that v0∗ and vω∗ are step functions. Next, we
show that in problem 2 with v0 = v0∗ , vω = vω∗ , and (m0 , mω ) = (1, m), the
implied seller’s payoff is vω∗ (m). For m = i, let (c, m0 ) be a solution of the
problem. Because [v0∗ (i + 1) − v0∗ (i)] ≤ D < ω/β, (7) is binding. Therefore,
vω∗ (i) = u(ω) + βv0∗ (i). For m ∈ [i, i + 1), this follows again from the result
for integers and the fact that v0∗ and vω∗ are step functions.
Thus, as one might expect, there is a degenerate steady state for the
above model. However, because the value functions are not concave, the
steady state may not survive lotteries in meetings. If lotteries are allowed,
then the constructed steady state survives under an additional restriction:
the buyer must want to offer the amount 1 with probability
1, rather than
u0 (c∗ )
with some lower probability. That is the case if u0 (ω−c∗ ) ≥ β (if c∗ is no
greater than the competitive equilibrium consumption in footnote 1). That
restriction may or may not hold. It does not hold if β is √
close enough to 1
because c∗ → ω as β → 1. It holds, for example, if u(c) = c and β 2 ≤ 1/3.
3.2
Application of the refinement
Now we apply the commodity-money refinement. For this model, we say
that an = 0 steady state (v0 , vω , π0 , πω ) is the limit of commodity-money
steady states (v0 , vω , π0 , πω ) as → 0 if (i) (π 0 , π ω ) converges weakly
to (π 0 , π ω ) and (ii) (v0 , vω ) converges uniformly to (v0 , vω ) over [0, 1] and
converges pointwise over (1, ∞).2 We say that an = 0 steady state satisfies
the refinement if it is such a limit. The next proposition shows that no = 0
degenerate monetary steady state satisfies the refinement.
Proposition 3 If (v0 , vω , π 0 , π ω ) is an = 0 monetary steady state that is
degenerate, then (v0 , vω , π 0 , π ω ) does not satisfy the commodity-money refinement.
2
We suspect that pointwise convergence of value functions is sufficient, but our proof
uses the stronger notion.
10
Proof. Assume by contradiction that (v0 , vω , π 0 , π ω ) satisfies the refinement and let {(v0 , vω , π 0 , π ω )} be the commodity-money steady states
whose limit is (v0 , vω , π 0 , π ω ).
First, we claim that for sufficiently small , π ω (0) > 0.5. To see why,
notice that for sufficiently small , π ω [0, 0.1] is sufficiently close to 1. Now if
πω (0) ≤ 0.5, then there exists l ∈ (0, 0.1) with π ω [l, 0.1] sufficiently close to
0.5. But now consider v0 (1 − l). It is feasible for a buyer with 1 − l to offer
1 − m to a seller with m ∈ [l, 0.1]. By convergence, v0 (1) is close to v0 (1)
and v0 (1 − l) is close to v0 (1 − l), and by proposition 1, v0 (1 − l) = v0 (0).
But then, such an offer implies that v0 (1 − l) is not close to v0 (1 − l), a
contradiction.
Now, let d be the unique positive solution to the equation x = 0.5u[f(x)].
Notice that the properties of f and the lower bound on u0 (0) imply that d
exists and that d < D ≡ v0 (1) − v0 (0). Now fix m ∈ (0, 1) and let be such
that (v0 , vω , π 0 , π ω ) satisfies π ω (0) > 0.5 and v0 (m) − v0 (0) < d. A lower
bound on v0 (m) is obtained by considering the following two feasible actions
for a buyer with m: use up all of m or offer m to a seller with zero. The
second action induces f [v0 (m) − v0 (0)] amount of the good from the seller
because v0 (m) − v0 (0) < d < D < u(ω)/β. Then those two actions for the
buyer give us the following inequality:
v0 (m) − βvω (0) ≥ max{ γ(m), 0.5u[f (v0 (m) − v0 (0))]}.
(18)
Let x ≡ v0 (m) − v0 (0). By βvω (0) = v0 (0), we can rewrite (18) as
x ≥ max{ γ(m), 0.5u[f(x)]}.
(19)
But (19) implies x ≥ d, a contradiction.
3.3
A full-support steady state
With two added technical assumptions, a bound on u0 (0) and a bound on
individual holdings of money, it can be shown that the model with = 0
has a full-support steady state with a strictly increasing and strictly concave
value function. Zhu [6] used those assumptions in a closely related model
to establish existence of such a steady state, and the arguments, although
lengthy, can be adapted to the model of this section. To show that such a
steady state satisfies the refinement, we proceed as follows.
As a sequence of steady states, we take the constant sequence, each term
of which is the above = 0 full-support steady state on [0, B], where B is the
11
bound on individual holdings. The results in [6] also imply that the steady
state value functions have positive left derivatives at B.3 We have to show
that this = 0 steady state is a steady state for all sufficiently small . That
is, we have to show that if is small, then no money is used up, or that for
all ω ≥ x and s ∈ {0, ω},
β[vs (ω) − vs (ω − x)] ≥ γ(x).
(20)
β[vs (ω) − vs (ω − x)] ≥ βvs0 (B)x ≥ γ 0 (0)x ≥ γ(x),
(21)
But
where vs0 (B) is the left derivative of vs at B and where the first inequality
follows from concavity of vs , the second holds for ≤ βvs0 (B)/γ 0 (0), and the
last follows from concavity of γ. Therefore, (20) holds for sufficiently small
.
Notice that the bound B plays a crucial role in showing that no one
uses up money. If we had adopted the money-as-jewelry specification, then
preservation of the stock of money would not arise as an issue. However,
even for that specification, there is no existence proof for B infinite.
4
Random Matching with Indivisible Money
Time is discrete. There is a [0, 1] continuum of each of N ≥ 3 types of
infinitely lived people, and there are N distinct produced and perishable
types of divisible goods at each date. A type n person, n ∈ {1, 2, ..., N},
produces only good n and consumes only good n+1 (modulo N). Each person
maximizes expected discounted utility with discount factor β ∈ (0, 1). For a
type n person, utility in a period is u(qn+1 ) − qn , where qn+1 is the amount of
good n + 1 consumed and qn is the amount of good n produced. The utility
function u : R+ → R+ is strictly increasing, strictly concave, continuously
differentiable and satisfies u(0) = 0, u0 (∞) = 0, and u0 (0) = ∞.
There exists a fixed stock of money. Let the average money holding per
type be denoted by m, the (smallest) unit of money by ∆(> 0), and the
exogenous (finite) upper bound on individual money holdings by B. As in
Zhu [5], there are lower bounds on m/∆ and B/m, and B/∆ is an integer.
Let B∆ = {0, ∆, 2∆, ..., B} denote the set of possible individual holdings of
money.
3
A proof is available upon request.
12
In each period, people are randomly matched in pairs. A meeting between
a type n person, a seller, and a type n + 1 person, a buyer, is called a singlecoincidence meeting. Other meetings are not relevant. In meetings, as above,
types and money holdings are observable, but any other information about
a person’s trading history is private.
A symmetric steady state is a value function, v, and a measure, π, both
defined on B∆ that pertain to the start of a date prior to meetings. The
choice problem of a buyer with money mb who meets a seller with money ms
is identical to that described by problem 1 except that choices are limited
to B∆ and imply holdings in B∆ .4 We let gb (mb , ms ) and gs (mb , ms ) be the
implied payoffs to the buyer and seller, respectively, and let P (mb , ms ) be
the set of maximizers for the buyer’s post-trade amount of money. Then the
value function satisfies
X
π(m0 )[gb (m, m0 ) + gs (m0 , m)].
(22)
v(m) = [N − (N − 2)β]−1
m0
We allow all possible randomizations over the elements of P (mb , ms ). Let
Λ(mb , ms ), a set of measures on B∆ , be given by
/ P (mb , ms )}.
Λ(mb , ms ) = {λ(.; mb , ms ) : λ(m; mb , ms ) = 0 if m ∈
Then π satisfies
1=
and
π(m) =
X
π(m)m,
(23)
(24)
1P
π(mb )π(ms )[λ(m; mb , ms )
2 mb ,ms
+ λ(mb + ms − m; mb , ms )]
(25)
/ B∆ in
for some λ(.; mb , ms ) ∈ Λ(mb , ms ). (As a convention, λ(z) = 0 if z ∈
(25).) Then, we say that (v, π) is a steady state if it satisfies (22)-(25).
4
As this suggests, we are here requiring that offers be deterministic. Lotteries are
discussed below.
13
4.1
Fiat monetary steady states
For = 0, Zhu [5] shows that there exists a steady state with v strictly
increasing and strictly concave, with v(0) = 0, and with π having full support.
Let us call any such steady state a nice steady state. Because the value
function is strictly increasing and concave, any nice steady state satisfies the
refinement.5
Existence of such nice steady states implies existence of non full-support
steady states. We first describe them and then show that none satisfy the
refinement. The model has three exogenous nominal quantities, (∆, m, B) ≡
z. Let k be an integer that exceeds unity and consider z 0 = kz and z 00 =
(∆, km, kB). As might be expected, neutrality holds for the comparison
between z and for z 0 , neutrality in the sense of real allocations. What about
the comparison between z and z 00 ? In terms of real allocations, z 00 has strictly
more steady states. In particular, no nice steady state for z 00 has the same
real allocation as any steady state for z. However, any steady state for z has
an equivalent (in terms of real allocations) steady state for z 00 . An equivalent
steady state is produced by replacing the trade of m units of money under z
by a trade of km units under z 00 and by letting (v 00 , π 00 ), the equivalent steady
state for z 00 , be given as follows: for each m ∈ B∆ , let π00 (km) = π(m), and
let v00 (km) = v(m) and v 00 (km + x) = v00 (km) for x = ∆, ∆ + 1, ..., k∆ − ∆.
That is, v00 is a step function with jumps at km. All of these claims are
proved in Zhu [5].6
This comparison between steady states for z and for z 00 raises the following
question. Under z 00 , are we likely to see a nice steady state implying nonneutrality relative to z or might we see a steady state which is equivalent to
one for z? We answer that by applying the refinement.
4.2
Application of the refinement
As above, an = 0 steady state satisfies the refinement if there exists a sequence of commodity-money steady states that converges to it. Here, because
5
The argument is a simpler version of that given for the model of alternating endowments.
6
As noted above, Zhu [5] proves existence of a nice steady state for a version without
lotteries. However, his arguments can be applied almost without change to a version with
lotteries. And such existence implies existence of non full-support steady states via the
same construction used in the text. In contrast to the model with divisible money, lotteries
do not impose any additional restrictions for such existence.
14
steady states are finite dimensional, convergence is defined in the usual way.
We show that the refinement rules out steady-state value functions which are
not strictly increasing. We prove this in two steps.
The first proposition shows that if (v, π) is an = 0 steady state and v
is not strictly increasing, then v is not strictly increasing at the lower end of
the support of π.
Proposition 4 Let (v, π) be an = 0 steady state. Let a = min{m : π(m) >
0}. (i) v(a) = 0. (ii) If a > 0, then v(a + ∆) = 0. (iii) If v is not strictly
increasing, then v(∆) = 0.
Proof. (i) If v(a) > 0, then a buyer with a has to trade with someone.
That gives rise to an inflow into holdings less than a, a contradiction.
(ii) Assume by contradiction that v(a + ∆) > 0. Because a buyer with a
can offer ∆ to a seller with a, it follows that v(a) > 0, which contradicts (i).
(iii) Assume by contradiction that v(∆) > 0. Let m = min{m0 : v(m0 ) =
v(m0 + ∆)}. It follows that m > 0 and v(m) > 0. Hence, the buyer with m
must trade with some probability. That is, there is a positive probability that
the buyer with m makes an offer p ≥ ∆ to some sellers. A buyer with m + ∆
has the same probability of meeting those sellers and can emulate the offers
made by the buyer with m. If so, then the buyer with m + ∆ ends up with
m+∆−p and the buyer with m ends up with m−p with positive probability.
But the definition of m and p ≥ ∆ imply v(m + ∆ − p) > v(m − p). That,
in turn, implies v(m + ∆) > v(m), a contradiction.
Now we apply the refinement.
Proposition 5 If (v, π) is an = 0 monetary steady state with v not strictly
increasing, then (v, π) does not satisfy the commodity-money refinement.
Proof. Assume by contradiction that (v, π) satisfies the refinement. As
in proposition 4, let a = min{m : π(m) > 0}. Let d denote the unique
positive solution to the equation x = π(a)
u(βx). By the definition of the
2N
refinement, there exists a steady state (v , π ) with v close to v and
X
x<a
π (a) <
1 π(a) 2
[
],
2N 2
and
π (a) >
15
π(a)
.
2
(26)
(27)
P
Notice that (26) must hold because by the definition of a, x<a π(x) = 0.
(The proof is written as if a > 0. As noted below, a = 0 is a simple special
case that does not make use of (26).) Now, in the (v , π ) steady state, a buyer
with a either trades with a seller with a or not. If a trades with a, then (27)
implies that the inflow into holdings less than a is bounded below by N1 [ π(a)
]2 .
2
P
π(a)
1
But then x<a π (x) > 2N
[ 2 ]2 , which contradicts (26). Therefore, a does
not trade with a. (If a = 0, this is the only possibility.) And because (v , π )
is a steady state, the person with a does not use up any money.
Now, consider someone with a + ∆. Among the feasible actions for this
person are two: use up ∆ and offer ∆ when meeting a seller with a. Because
someone with a does not use up any money and does not trade with a seller
with a, those feasible actions imply the following inequality:
v (a + ∆) − v (a) ≥ max{ γ(∆),
π(a)
u[βv (a + ∆) − βv (a)]}.
2N
But this implies v (a + ∆) − v (a) ≥ d. However, because proposition 4
implies that v(a + ∆) − v(a) = 0, the last inequality violates the condition
that v is close to v.
We have not proved that any steady state that satisfies the refinement
has full support. We have shown that a non full-support steady state that
satisfies the refinement cannot have a step-function value function. And we
know that it cannot have a concave value function because Zhu [5] shows that
concavity of a steady state value function implies that the accompanying π
has full support. We have not ruled out a non full-support steady state with
a strictly increasing value function that is not concave.
5
Concluding Remarks
Our results show that the commodity-money refinement has bite in some
models of pairwise trade. As in most other models, the equilibria which do
not satisfy the refinement are those in which the marginal value of money is
zero. An exception is Zhou [4], in which an equilibrium with a step-function
value function satisfies the refinement.7 In any case, if monetary objects do,
indeed, have some value as commodities, then the refinement should always
be applied.
7
In Zhou’s model, the good is available in one indivisible unit, money is divisible,
holdings of money are private information, and sellers post a price of one unit of the good.
16
There are, of course, other refinements that might be considered for steady
states of the models we have studied. An obvious one, which perhaps should
not even be called a refinement, is local stability. For now, we have no results
for other refinements or for local stability.
Throughout, we have assumed take-it-or-leave-it offers by buyers because
existence of a full-support steady state with a strictly increasing and concave
value function has been established only for that bargaining outcome.8 That
being the case, it seems premature to discuss a refinement for versions with
other bargaining outcomes.
References
[1] T. Bewley, The optimum quantity of money, in Models of Monetary
Economies, (J. Kareken and N. Wallace, Eds.) pp. 169-210, Federal Reserve Bank of Minneapolis, Minneapolis, 1980.
[2] S. Shi, Money and prices: a model of search and bargaining, J. of Econ.
Theory 67 (1995) 467-498.
[3] A. Trejos, and R. Wright, Search, bargaining, money and prices, J. of
Political Economy 103 (1995) 118-141.
[4] R. Zhou, Does commodity money eliminate the indeterminacy of equilibria?, J. of Econ. Theory 110 (2003) 176-190.
[5] T. Zhu, Existence of a monetary steady state in a matching model: indivisible money, forthcoming, J. of Econ. Theory.
[6] T. Zhu, Existence of a monetary steady state in a matching model: divisible money, miemo, Cornell University, 2003.
8
The mapping studied to establish existence preserves concavity of value functions
under take-it-or-leave-it offers, but not under other bargaining outcomes. That is why
Zhu’s [5] approach does not apply directly under other bargaining outcomes.
17